# Analytic approximation to delayed convection dominated systems through transforms

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## Abstract

In this paper we consider a family of singularly perturbed delay differential equation of convection diffusion type. When the perturbation parameter is very small, the solution of the problem exhibits layer behavior. In the layer region the solution changes rapidly, while away from this region the change in the solution is moderate. This simultaneous presence of two different scales phenomena makes the problem stiff. In this work, the problem is solved by applying a new Liouville–Green transform and the asymptotic solutions are obtained. Application to multi-point boundary value problem is also illustrated. Several test examples are taken into account so as to test the efficiency of the proposed method. The method presented is compared with other existing numerical or asymptotic methods. It is observed that the method presented is very easy to implement and is capable of reducing the size of calculations significantly while still maintaining high accuracy of the solution.

## Keywords

Asymptotic solution Liouville Green transform Layer behavior Delay differential equation## References

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